

from __future__ import annotations

from collections.abc import Callable
from typing import Generic, TypeVar

__A			:					str						=			TypeVar('T')
__A			:					Optional[int]						=			TypeVar('U')





class 			_SCREAMING_SNAKE_CASE (   Generic[T, U]		):







   '''simple docstring'''




   def __init__(       self					:			Any     , __lowerCamelCase					:			T | None     , __lowerCamelCase					:			U | None			):
       SCREAMING_SNAKE_CASE										=   key
       SCREAMING_SNAKE_CASE										=   val
       SCREAMING_SNAKE_CASE										=   None
       SCREAMING_SNAKE_CASE										=   None





   def __repr__(       self					:			int			):
       return (
           f"Node: key: {self.key}, val: {self.val}, "
           f"has next: {bool(self.next			)}, has prev: {bool(self.prev			)}"
       )





class 			_SCREAMING_SNAKE_CASE (   Generic[T, U]		):







   '''simple docstring'''




   def __init__(       self					:			Tuple			):
       SCREAMING_SNAKE_CASE										=   DoubleLinkedListNode(UpperCamelCase__     , UpperCamelCase__			)
       SCREAMING_SNAKE_CASE										=   DoubleLinkedListNode(UpperCamelCase__     , UpperCamelCase__			)
       SCREAMING_SNAKE_CASE							,	SCREAMING_SNAKE_CASE										=   self.rear, self.head





   def __repr__(       self					:			Optional[int]			):
       SCREAMING_SNAKE_CASE										=   ["DoubleLinkedList"]
       SCREAMING_SNAKE_CASE										=   self.head
       while node.next is not None:
           rep.append(str(UpperCamelCase__			)			)
           SCREAMING_SNAKE_CASE										=   node.next
       rep.append(str(self.rear			)			)
       return ",\n    ".join(UpperCamelCase__			)





   def      _snake_case		(       self					:			Union[str, Any]     , __lowerCamelCase					:			DoubleLinkedListNode[T, U]			):
       SCREAMING_SNAKE_CASE										=   self.rear.prev

       # All nodes other than self.head are guaranteed to have non-None previous
       assert previous is not None

       SCREAMING_SNAKE_CASE										=   node
       SCREAMING_SNAKE_CASE										=   previous
       SCREAMING_SNAKE_CASE										=   node
       SCREAMING_SNAKE_CASE										=   self.rear





   def      _snake_case		(       self					:			str     , __lowerCamelCase					:			DoubleLinkedListNode[T, U]			):






       if node.prev is None or node.next is None:
           return None

       SCREAMING_SNAKE_CASE										=   node.next
       SCREAMING_SNAKE_CASE										=   node.prev
       SCREAMING_SNAKE_CASE										=   None
       SCREAMING_SNAKE_CASE										=   None
       return node





class 			_SCREAMING_SNAKE_CASE (   Generic[T, U]		):







   '''simple docstring'''




   lowerCamelCase__				=      {}





   def __init__(       self					:			Optional[Any]     , __lowerCamelCase					:			int			):
       SCREAMING_SNAKE_CASE										=   DoubleLinkedList()
       SCREAMING_SNAKE_CASE										=   capacity
       SCREAMING_SNAKE_CASE										=   0
       SCREAMING_SNAKE_CASE										=   0
       SCREAMING_SNAKE_CASE										=   0
       SCREAMING_SNAKE_CASE										=   {}





   def __repr__(       self					:			List[str]			):
       return (
           f"CacheInfo(hits={self.hits}, misses={self.miss}, "
           f"capacity={self.capacity}, current size={self.num_keys})"
       )





   def __contains__(       self					:			str     , __lowerCamelCase					:			T			):
       return key in self.cache





   def      _snake_case		(       self					:			Union[str, Any]     , __lowerCamelCase					:			T			):






       if key in self.cache:
           self.hits += 1
           SCREAMING_SNAKE_CASE										=   self.cache[key]
           SCREAMING_SNAKE_CASE										=   self.list.remove(self.cache[key]			)
           assert node == value_node

           # node is guaranteed not None because it is in self.cache
           assert node is not None
           self.list.add(UpperCamelCase__			)
           return node.val
       self.miss += 1
       return None





   def      _snake_case		(       self					:			Optional[Any]     , __lowerCamelCase					:			T     , __lowerCamelCase					:			U			):






       if key not in self.cache:
           if self.num_keys >= self.capacity:
               # delete first node (oldest) when over capacity
               SCREAMING_SNAKE_CASE										=   self.list.head.next

               # guaranteed to have a non-None first node when num_keys > 0
               # explain to type checker via assertions
               assert first_node is not None
               assert first_node.key is not None
               assert (
                   self.list.remove(UpperCamelCase__			) is not None
               )  # node guaranteed to be in list assert node.key is not None

               del self.cache[first_node.key]
               self.num_keys -= 1
           SCREAMING_SNAKE_CASE										=   DoubleLinkedListNode(UpperCamelCase__     , UpperCamelCase__			)
           self.list.add(self.cache[key]			)
           self.num_keys += 1

       else:
           # bump node to the end of the list, update value
           SCREAMING_SNAKE_CASE										=   self.list.remove(self.cache[key]			)
           assert node is not None  # node guaranteed to be in list
           SCREAMING_SNAKE_CASE										=   value
           self.list.add(UpperCamelCase__			)





   @classmethod
   def      _snake_case		(       cls					:			List[Any]     , __lowerCamelCase					:			int = 128			):






       def cache_decorator_inner(__lowerCamelCase					:			Callable[[T], U]			) -> Callable[..., U]:
           def cache_decorator_wrapper(*__lowerCamelCase					:			T			) -> U:
               if func not in cls.decorator_function_to_instance_map:
                   SCREAMING_SNAKE_CASE										=   LRUCache(UpperCamelCase__			)

               SCREAMING_SNAKE_CASE										=   cls.decorator_function_to_instance_map[func].get(args[0]			)
               if result is None:
                   SCREAMING_SNAKE_CASE										=   func(*UpperCamelCase__			)
                   cls.decorator_function_to_instance_map[func].put(args[0]     , UpperCamelCase__			)
               return result

           def cache_info() -> LRUCache[T, U]:
               return cls.decorator_function_to_instance_map[func]

           setattr(UpperCamelCase__     , "cache_info"     , UpperCamelCase__			)  # noqa: B010

           return cache_decorator_wrapper

       return cache_decorator_inner


if __name__ == "__main__":
    import doctest

    doctest.testmod()

import math

def a__       (			A_   ):






  '''simple docstring'''





  if 1 < number < 4:
    # 2 and 3 are primes
    return True
  elif number < 2 or number % 2 == 0 or number % 3 == 0:
    # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
    return False

  # All primes number are in format of 6k +/- 1
  for i in range(5,	int(math.sqrt(A_   ) + 1   ),	6   ):
    if number % i == 0 or number % (i + 2) == 0:
      return False
  return True

def a__       (			A_ = 0.1   ):






  '''simple docstring'''





  __magic_name__								=			3
  __magic_name__								=			3

  while primes / (2 * j - 1) >= ratio:
    for i in range(j * j + j + 1,	(j + 2) * (j + 2),	j + 1   ):
      primes += is_prime(A_   )
    j += 2
  return j


if __name__ == "__main__":
    import doctest

    doctest.testmod()

